Problems & Puzzles: Puzzles

Puzzle 1071 Follow-up to Puzzle 1070 Emmanuel Vantieghem a sent solution to Q4 from Puzzle 1070 using just once each integer of a subset of all the integers from 1 to 118. But he added:  "I'm almost sure there must be bigger solutions." This is the palprime to beat: 99.8.97.96.95.94.93.92.91.87.86.85.84.83.82.81.76.75.74.73.72.71.70.1.65.64.63.62.61.54.53.52.51.32.31. 43.42.41.21.60.11.106.12.14.24.34.13.23.15.25.35.45.16.26.36.46.56.10.7.17.27.37.47.57.67.18.28.38.48. 58.68.78.19.29.39.49.59.69.79.89.9, Integers Used=81, Integers not used = 118-81= 37 Here are the integers not used: 2, 3, 4, 5, 6, 20, 22, 30, 33, 40, 44, 50, 55, 66, 77, 80, 88, 90, 98, 100, 101, 102, 103, 104, 105, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118 The palprime without dots: 99897969594939291878685848382817675747372717016564636261545352513231434241216011106121424 3413231525354516263646561071727374757671828384858687819293949596979899 (159 digits, palindrome-ness and primality checked by CR) Q. Find the largest palprime using the Vantieghem approach. Note: Please send your solutions separating each atomic number by a dot.

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During the week 8-14 Jan, 2022, contributions came from Emmanuel Vantieghem, Michael Hürter

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Emmanuel wrote:

This is my 'biggest' configuration (molecule ?) of  96 different chemical elements :

99.8.97.96.95.94.93.92.91.90.1.87.86.85.84.83.82.81.76.75.74.73.72.71.

17.65.64.63.62.61.16.54.53.52.51.15.43.42.32.31.13.41.14.21.12.80.11.110.

77.70.111.108.2.112.4.114.3.113.23.24.34.5.115.25.35.45.6.116.26.36.46.56.

7.117.27.37.47.57.67.18.28.38.48.58.68.78.109.19.29.39.49.59.69.79.89.9

Concatenation shows the palprime:

998979695949392919018786858483828176757473727117656463626116545352511543423231134114211280111107770
1111082112411431132324345115253545611626364656711727374757671828384858687810919293949596979899 (193 digits). (Primality and Palindrome-ness checked by CR)

 

Unused atomic numbers :
10, 20, 22, 30, 33, 40, 44, 50, 55, 60, 66, 88, 98, 100, 101, 102, 103, 104, 105, 106, 107,118.

But I think it is not maximal.

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Michael wrote:

I found the following solution:

 

1.118.2.116.3.112.20.107.4.117.5.103.95.21.93.109.88.97.84.101.115.86.73.71.69.77.65.76.62.83.64.58.74.23.52.99.55.46.94.49.85.
39.18.47.56.32.41.38.26.108.43.40.106.72.111.27.60.104.34.80.16.28.31.42.36.57.48.19.35.89.44.96.45.59.92.53.24.78.54.63.82.66.
75.67.79.6.17.37.68.51.110.14.87.98.8.90.13.91.25.9.30.15.7.114.70.10.22.113.61.12.81.11

111821163112201074117510395219310988978410111586737169776576628364587423529955469449853918475632413826108434010672111276010
434801628314236574819358944964559925324785463826675677961737685111014879889013912593015711470102211361128111
(Quantity of digits 231, Primality and Palindrome-ness, checked by CR)
 

The integers not used are:
29, 33, 50, 100, 102, 105

"In your opinion, or according to your approach, can you assure is this the biggest solution?" (CR)

This is a difficult question, i am not sure, but:

 

I think there are better solutions.

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Dmitry Kamenetsky wrote on March 23, 2022:

I have an improvement for Puzzle 1071. First I realized that 100 cannot be in the sequence as it has no match in the palindrome. However, removing just 100 is not sufficient to construct a palindrome. We also need to remove one of 10, 19, 90, 91 or 109 to make the digits work. I chose to remove 10. I then found palindromes with the remaining numbers and checked their primality. It took me about 100 attempts until I finally found a palindrome that is prime. It has 241 digits. Here it is:

 

37.90.116.85.21.59.54.75.39.114.4.31.14.2.35.34.48.94.3.101.104.67.30.18.96.19.62.88.22.111.42.78.5.63.36.27.71.9.20.118.
51.97.80.106.70.1.83.46.52.92.66.55.117.105.23.68.74.7.89.|9|9.87.47.86.32.50.17.115.56.6.29.25.64.38.107.60.108.79.15.
81.102.91.77.26.33.65.8.72.41.112.28.82.69.16.98.103.76.40.110.13.49.84.43.53.24.113.44.11.93.57.45.95.12.58.61.109.73.

 

or 

37901168521595475391144311423534489431011046730189619628822111427856336277192011851978010670183465292665511
7105236874789|9|9874786325017115566292564381076010879158110291772633658724111228826916981037640110134984435
32411344119357459512586110973 (Primality and Palindromeness checked by CR; 241 digits; two unused integers: 100 and 10.
 

-Can it be another larger palprime with the same quantity of digits for this puzzle? (CR)

-Yes that's possible and quite likely (DK)

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